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1

Network connectivity dynamic modelling

Eid Mohamed

Safety and Reliability of Systems and Processes

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2021

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Summer Safety and Reliability Seminar 2021

117--127

EN

The chapter presents an approach based on basic notions issued from the graph theory and from the system reliability theory. The approach seeks to describe the transitions in the connectivity state of non-directional graphs induced by systemic losses of nodes and edges. Each component loss (node or edge) represents a transition and results in a connectivity degradation. Some of the transitions are classified as non-critical while others are critical. Degradations are measured using the notion of topological graph diameter issued from the graph theory. The critical transition notion is issued from the system reliability theory. The approach determines the degraded graph diameter corresponding to each possible transition and subsequently the criticality of the transition. The criticality threshold is determined by the highest acceptable connectivity order which is a function of the degraded graph diameter. A network with 9 nodes and 15 edges is used as an academic study case to illustrate the applicability of the approach. Nodes are supposed to be identical, as well as the edges. All network components (nodes and edges) are mutually independent. These precedent hypotheses are intended to evacuate all sources of numerical useless complexity. As our main objective is to highlight the original characteristic of the proposed approach.

2

Availability of WDM Multi Ring Networks

Rados I., Rados K.

Journal of Telecommunications and Information Technology

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2015

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nr 1

53--58

EN

This paper presents a new approach to modeling the availability of the networks composed of multiple interconnected rings in two nodes. For availability modeling method algebraic formulation is used. Using this method, through the availability of multiple ring networks consisting of two and three-rings connected to two nodes, a general expression for the availability between two terminals of multiple ring networks is derived. To perform the expression some real assumptions were taken and the analytical calculation showed that the use of these expressions under these assumptions provides real values for availability between the two terminals of multiple ring networks. Information on the availability of links and nodes is taken from previously published works.

3

Lyapunov, Sylvester and Riccati equations with some applications

Kaczorek T.

Pomiary Automatyka Kontrola

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2007

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R. 53, nr 6

63-67

EN

An overview of the differential and algebraic Lyapunov, Sylvester and Riccati equations is presented. A special attentions is focused on relationship between the equations and their applications in control systems theory. The well-known classical Cayley-Hammilton theorem is extended for the Lyapunov time-varying systems.

PL

W pracy podano przegląd różniczkowych i algebraicznych równań Lapunowa, Sylvestera i Riccatiego. Szczególną uwagę zwrócono na związki występujące miedzy tymi równaniami oraz ich zastosowaniami w teorii sterowania i systemów. Uogólniono klasyczne twierdzenie Cayleya - Hamiltona na układy Lapunowa o zmiennych w czasie parametrach.